dimensioning of silicone adhesive joints: eurocode
TRANSCRIPT
Glass Struct. Eng.https://doi.org/10.1007/s40940-020-00128-4
SPECIAL ISSUE CHALLENGING GLASS
Dimensioning of silicone adhesive joints: Eurocode-compliant, mesh-independent approach using the FEM
Micheal Drass · Michael A. Kraus
Received: 1 February 2020 / Accepted: 10 July 2020© The Author(s) 2020
Abstract This paper deals with the application ofthe semi-probabilistic design concept (level I, DIN EN1990) to structural silicone adhesives in order to cali-brate partial material safety factors for a stretch-basedlimit state equation. Based on the current legal situationfor the application of structural sealants in façades, anew Eurocode-compliant design concept is introducedand compared to existing design codes (ETAG 002).This is followed by some background information onsemi-probabilistic reliability modeling and the gen-eral framework of the Eurocode for the derivation ofpartial material safety factors at Level I. Within thispaper, a specific partial material safety factor is derivedfor DOWSIL 993 silicone on the basis of experimen-tal data. The data were then further evaluated undera stretch-based limit state function to obtain a partial
M. Drass (B) · M. A. KrausM&M Network-Ing UG (haftungsbeschränkt),Lennebergstraße 40, 55124 Mainz, Germanye-mail: [email protected];[email protected]://www.mm-network-ing.de
M. DrassInstitute for Structural Mechanics and Design, TechnischeUniversität Darmstadt TU Darmstadt, Franziska-Braun-Str.3, 64287 Darmstadt, Germany
M. A. KrausCivil and Environmental Engineering, Stanford University,Y2E2, 473 Via Ortega, Stanford, CA 94305, USAe-mail: [email protected];[email protected]://www.mm-network-ing.de
material safety factor for that specific limit state func-tion. This safety factor is then extended to the appli-cation in finite element calculation programs in sucha way that it is possible for the first time to performmesh-independent static calculations of silicone adhe-sive joints. This procedure thus allows for great opti-mization of structural sealant design with potentiallyhigh economical as well as sustainability benefits. Anexample for the static verification of a bonded façadeconstruction by means of finite element calculationshows (i) the application of EC 0 to silicone adhesivesand (ii) the transfer of the EC 0 method to the finiteelement method with the result that mesh-independentultimate loads can be determined.
Keywords Partial material safety factor · Structuralsilicone adhesive · SSG façades · Design andcomputation
1 Introduction and current situation
State-of-the-art glass façades are designedwith a strongemphasis on a transparent appearance with minimalvisibility of the supporting structures. During the lastfifty years a lot of experience with structural siliconeadhesive joints in façade design has been gained world-wide. Beginning with linear adhesive joints, which areused along a window system for homogeneous loadtransfer (Staudt et al. 2018), up to local fixings, whereglass panes are only bonded locallywith so-called point
123
M. Drass, M. A. Kraus
fixings (Drass et al. 2019b; Santarsiero and Louter2019). More recent developments deal with so-calledlaminated joints, in which either a puck is laminatedinto a laminated safety glass (LSG) or something islaminated onto a glass (Bedon and Santarsiero 2018).
For the dimensioning of silicone adhesive joints infaçades, there are two standards, ETAG 002 (2012)and ASTM C1401 (2002), which are common prac-tice throughout the world. Both calculation methodsare based on a linear analysis of the geometrical andmaterial behavior and assume an even load distribution.Furthermore, a constant stress state of the adhesive isassumed, which leads to a nominal stress analysis. Inorder to ensure sufficient redundancy or safety in thedesign of the silicone adhesive joint, these two stan-dards use aglobal safety concept, so thatmodeling inac-curacies (load and constitution behavior), temperature,humidity and aging effects (salt, detergent, SO2, UV)are covered.Therefore, a global safety factor ofγtot = 6is introduced to achieve a sufficiently large safety mar-gin (ETAG 002 2012). Regretfully, the exact history ofthe determination of the global security factor γtot can-not be reconstructed at this time based on the currentversion of ETAG 002 (2012). Therefore, a discussionabout the safety factor has been sparked in the industrytoday and the demand for a comprehensible calculationof a correct and justifiable safety factor has arisen.
Given that little work is currently being done on themethodologically correct and thusEurocode-compliantderivation of a partial safety factor for silicone adhe-sives in the façade sector, this contribution deals on theone hand with the development and presentation of aEurocode-compliant partial safety factor, the discus-sion of the influence of potential limit state functionsand on the other hand with the implementation of aLevel I approximation of a partial safety factor for thesilicone adhesiveDOWSIL993. The proposedmethod-ology is generally valid, so that a partial safety factorcan also be derived for other structural silicones underdifferent limit state functions. The described method isbased on the calibration procedure given in EurocodeEC 0 and additionally uses test results and modelingcontent fromETAG002 (2012) to provide a simple linkbetween the two concepts. With the help of the deter-mined partial safety factor, the connection between thetwo concepts is easily established and it is possible todesign and calculate silicone adhesive joints accordingto the partial safety factor concept of DIN EN 1990Eurocode (2010). The paper concludes with a practi-
cal example in which a bonded façade construction isdimensioned. As a highlight, the paper transfers thesafety factor to the Finite Element Method (FEM) withthe result that for the first time mesh-independent ulti-mate loads can be determined with the remark that onesimple and easy FE calculation has to be performedon the H-sample to calibrate the structural parameteraccording to Eurocode.
2 Historical survey of design philosophies
Historically, there are two types of design philosophieswith different safety concepts for the design of buildingcomponents in civil engineering:
– allowable stress design method with a global safetyconcept,
– limit state design method with a semi-probabilisticsafety concept.
In the following, both concepts are briefly presentedfor reasons of comprehensibility.
2.1 Allowable stress design method with global safetyconcept
Structural buildings have been designed in the pastunder the principle “the greater the uncertainty, thegreater the factor of safety must be for this struc-ture”. Accordingly, safety factors of magnitudes 1-6were chosen depending on the structure, uncertain-ties in loads etc. However, on the one hand therewas no normative regulation on how the safety fac-tors should be calculated accordingly to a standard andon the other hand, reliability analysis provided guid-ance for obtaining theoretically and methodically cor-rect safety factors for given levels of reliability butwith no formal governmental accreditation. Therefore,these safety factors always were determined based onthe experience of the respective engineers and design-ers. The reduction by a safety factor was related toa characteristic strength value of the material underinvestigation, which was determined from experimen-tal observations. The selection of the safety factor thendepended on the target reliability of the strength estima-tion. In the second half of the nineteenth century, thetheory of elasticity began to become accepted in thepractice of structural engineers. This is also the reasonwhy the design is carried out with linear calculations
123
Dimensioning of Silicone Adhesive Joints...
Fig. 1 Nonlinearforce-displacement behaviorof structures, joints ormaterials with (a) brittlefailure, (b) semi-ductilefailure and (c) ductilebehavior
(both material and geometrical) based on engineeringor nominal stresses, which is known as allowable stressdesign (ASD) method.
The ASD, using a global safety concept, calculatesthe maximum load or stress in the component for theentire life cycle. As a result, the limit state is reached onthe action side using linear elasticity theory. For struc-tural silicones, an engineering stress-based limit statefunction is commonly used. In the design, the calcu-lated maximum stress must be smaller than the charac-teristic strength (5% quantile) of the examinedmaterialreduced by a global safety factor. If this process is sum-marized in terms of formula one obtains
σ ≤ σdes = σlim
FS. (1)
In (1), the parameterσ represents the engineering stressin the material under maximum load for the whole life-time of the structure, σdes is the (engineering) designstrength, σlim the characteristic strength of the mate-rial and the abbreviation FS stands for factor of safety.Hence, the ASD approach is also applied in ETAG 002(2012), which regulates the design of silicone adhesivejoints.
Following the comments of Blockley (1992), theASDmethod can be summarized in the traditional wayas follows:
– Under service loads, all parts of the constructionbehave linearly elastic.
– In case the service loads have been calculated to beso high that the probability of exceeding them islow, and if the allowable stresses are chosen to be asufficiently small fraction of a limit stress, then thestructure has an excellent chance to have no damagewithin its lifetime.
This definition is also taken up by ETAG 002 (2012),which assumes linear elastic material behaviour for
reasons of simplicity. The ASD approach is character-ized by its simplicity and vividness, so that it quicklyfound its way into engineering practice and has estab-lished itself over several decades. It can also be said thatthis approach is conservative and therefore on the safeside. Finally, the ASD approach is a fully determin-istic approach in accordance to Marek and Kvedaras(1998).
However, there are also many disadvantages for theabove-mentioned concept from a scientific, probabilis-tic and economic point of view, which Blockley (1992)summarises as follows:
– Stress-strain relationship is not always linear, espe-cially not for structural silicone adhesives.
– Material non-linearities may occur due to timeeffects (creep and relaxation) which are disre-garded.
– Load effect and deformation are not always in alinear relationship.
– The material behavior beyond the linearity limitscan be ductile with load carrying capacity reserves.
– The probability of exceeding the limit state at thebeginning of non-linearity depends decisively onthe statistical properties of the loads, the materials,the idealizations used to create a calculationmodel,etc. Consequently, the reliability of the elementswithin the structure or the reliability of the differentstructures can significantly fluctuate.
To illustrate this and to address nonlinearities of struc-tures, joints or materials, Fig. 1 is an example of it.
Here, one can see that the load-deformation behav-iour past the theoretical limit of linear response may bebrittle, semi-ductile or ductile with very large reserve.In anASDapproach, however, the good-natured ductilebehaviour cannot be taken into account. Therefore, theASD approach in this example gives the same result for
123
M. Drass, M. A. Kraus
all three fracture behaviors, which can be classified asconservative.
Additionally, it is important to note at this point thatmany of the limitations mentioned are violated in theirgeneral validity by ETAG 002 (2012). If one takes acloser look at ETAG’s concept for the static dimen-sioning of silicone adhesive joints and generally thestructural behavior of silicone adhesives, they behavenon-linear-elastically (Drass et al. 2019a, b), are time-and rate-dependent (Kraus et al. 2017) and tend to creepunder permanent load (Botz et al. 2019). According tothe list of disadvantages defined by Blockley (1992),new concepts for any type of material have been devel-oped which circumvent the mentioned disadvantages.Here the so-called limit state design method has gainedacceptance. It no longer concentrates only on the ser-vice condition under full load and reduction of materialresistances, but deals with the limit of structural use-fulness.
2.2 Limit state design method with semi-probabilisticsafety concept
In the limit state design (LSD) method, the limitstrength, ultimate strength, collapse strength, maxi-mumcapacity of structures, columns, beams or connec-tions is calculated and then reduced to take into accountpossible influences from uncertainties in the strength ofmaterials, manufacturing aspects and uncertainties ofthe structures in the final state after construction. Thefactorized strength, or resistance side, is then evaluatedagainst the calculated load effect due to the correspond-ing maximum loads. The loads, also called the actionside, are then increased to take into account the uncer-tainties of the loads acting on the structure during itslifetime. In the evaluation of structural reliability, theconcept of a limit state surface has thus become estab-lished. Here, the multidimensional domain of randomvariables is divided into safe and uncertain domains(Marek and Kvedaras 1998).
The safety and reliability of buildings is on the onehand determined by the variability within the actionsand resistances of it and on the other hand by potentialerrors in planning, execution and use. Human miscon-duct however, cannot be detected, handled and coveredby a safety concept, but must be excluded as far aspossible by targeted measures such as checking of astructural design computation, quality assurance dur-
ing the construction process of the structure and main-tenance during use. Only the stochastic character ofthe input variables for actions and resistances can bedetermined by probabilistic methods. This requires aquantification of the stochastic uncertainties in actionsand resistances.
Basically, the core of the design philosophy in DINEN1990Eurocode (2010) is the solution of the inequal-ity
Ed ≤ Rd, (2)
with
Ed = γQ · Ek (3)
and
Rd = Rk
γM. (4)
In the above-mentioned equations, Ed represents thedesign value of an action and Rd the design value ofthe resistance, the indices d means design and k standsfor its characteristic value. To calculate the limit stateof the action side, the characteristic value Ek is multi-plied by a factor γQ, whereas the resistance side or thecharacteristic value Rk is divided by a partial factor γMcovering uncertainties in the resistance model.
Considering (2), both the action side and the resis-tance side are calculated separately as limit states andcompared against each other. The partial safety factorsfor established materials such as concrete, steel or tim-ber have been calibrated such, that a certain economi-cally acceptable target reliability (given in and definedby the national annexes of the Eurocode 0) is reached.In contrast to ASD, material and geometric nonliner-aties and imperfections as well are considered in theLSD approach which is a major advantage. The LSDapproach provides the same results or is identical toASD for the case that the end of the elastic reactionis the limit state, the structure is perfect and the mate-rial behaviour is ideally linear elastic. To illustrate thedifferent approaches once again, we will take a closerlook at the action side in the following. In the case ofLSD, the limit state of the action is calculated and it ischecked whether the actual load is below it. With ASD,on the other hand, the strength limit must be main-tainedunder the actual load.To illustrate the differencesbetween both approaches, Fig. 2 can be adduced.
It is quite clear that ASD ignores the non-linearityandmay lead to conservative results. An opposite effectcan occur in insulating glass under climatic loads,
123
Dimensioning of Silicone Adhesive Joints...
Fig. 2 Example of the twodesign methods of limitstate design and allowablestress design
where it can be on the unsafe side if the non-linearityof the shear bond, for example, is ignored.
The LSD approach can again be divided into twosubgroups. The semi-probabilistic approach calculatespartial safety factors on the impact and resistance sideto define both limit states. From the designer’s point ofview, this concept is still deterministic, but a probabilis-tic approach is found in the safety factors. Hence, thismethod is called the semi-probabilistic partial safetyfactor concept. The second approach deals with fullyprobabilistic methods for the implementation of reli-ability assessment of structures. This approach is notdescribed in detail in the following and requires greaterefforts in terms ofmodelling and evaluation of the prob-lem at hand. For other materials such as steel or con-crete, the semi-probabilistic approach is state of theart and has been introduced throughout Europe by theEC199x series and does not pose any difficulties forcivil engineers (DIN EN 1990 Eurocode 2010), espe-cially engineering education on universities since tenyears teach Bachelor and Master students this designphilosophy.
With regard to the construction industry, reliability isassessed by comparing the calculated reliability indexβ with the reliability index that is regarded as adequatefor the system under evaluation from previous experi-ence. For this purpose, onemust establish a relationshipbetween the capacity Rd (for example, the strength) ofthe system and the demand Ed (for example, the load)such that if capacity and demand are equal, there is alimiting state of interest. The margin of safety, definedas S = g(E, R) = R − E , is another example of thisstate, where S > 0 represents the safe state, S < 0the failure state. For reasons of completeness, S = 0defines the limiting state. Accordingly, the probabilityof failure p f is given by
p f = Pr[R − E ≤ 0] = Pr[S ≤ 0], (5)
where the Pr [•] defines any probability operatorapplied to the argument •. A fundamental measure ofreliability theory is the so-called reliability index β,which is defined as a measure of an assigned proba-bility of failure at a design point. The reliability indexis usually set to β = 3.8 for the ultimate limit stateand the permanent design situation for a design lifeof a building of 50 years. The reliability index can becalculated by the expected value μ and the varianceσ 2 under assumption of a Gaussian distribution for theaction and resistance side
β = μR − μE
σ 2R − σ 2
E
. (6)
The terms safety and reliability play a decisive rolein the construction industry. For example, the safetyconcepts currently used in construction are based ona semi-probabilistic safety concept with partial safetyfactors on the action and resistance side. Since theaction side is already described by the Eurocode, theaim of the present study is therefore to derive a partialmaterial safety factor for a structural silicone accord-ing to the Eurocode, so that a uniform design procedurefor bonded façade systems can be established. Figure 3gives a general and schematic overview of the par-tial safety factor concept according to DIN EN 1990Eurocode (2010). From this diagram, it is directly evi-dent how the partial safety factor influences the resis-tance side.
It is important to note that according to the Eurocodethere is a partial safety factor γm for a material or prod-uct property and a partial safety factor γM for a compo-nent property, taking into account model uncertaintiesand size deviations.
123
M. Drass, M. A. Kraus
Fig. 3 General overview of the partial factor system in the Eurocodes, from Gulvanessian et al. (2012a) and cf. Drass and Kraus (2020),Kraus and Drass (2020)
3 Calibration of partial safety factor
In the last section two different design philosophiesfor the design of building structures were presented ingeneral. In this section, the material resistance accord-ing to DIN EN 1990 Eurocode (2010) will be derivedspecifically for the structural silicone DOWSIL 993.
It shall be shown that on the one hand the derivationof a partial safety factor at a given limit state func-tion according to DIN EN 1990 Eurocode (2010) ispossible in a few lines of code. On the other hand,the LSD approach with the partial safety factor con-cept for structural silicones will be demonstrated forthe first time. The advantages are that the non-linearmaterial behavior of the silicone is taken into account,
123
Dimensioning of Silicone Adhesive Joints...
which leads to less conservative results. This allowsa better utilization of the silicone, i.e. the design ismore material-orientated. Furthermore, by using DINEN 1990 Eurocode (2010), bonded glass componentscan be verified with one and the same safety conceptwithoutmixing theASDandLSDapproach. This offersadvantages in the calculation of bonded façade compo-nents, since two different calculations (once ASD andonce LSD) do not have to be performed.
3.1 Mathematical framework: partial material safetyfactor
Following DIN EN 1990 Eurocode (2010); Sørensen(2002), the design values of material or product prop-erties X are determined by
Xd = ηXk
γm= η
γm{mX − knVX} . (7)
In this context, Xk represents the characteristic strengthvalue (5%quantile) andη is themean value of a conver-sion factor that reflects differences between thematerialstrength in the calculationmodel and in the actual struc-ture as well as laboratory size effects (humidity, tem-perature, scale and size effects, etc.). Typically, η = 1can be assumed, cf. DIN EN 1990 Eurocode (2010).However, since in this paper more concern is put onthe factor η, it is calculated or put in connection withthe ETAG 002 (2012). Returning to (7), the variablemX describes the mean value of the material propertyX for n samples, kn represents the fractile value for thecharacteristic value and VX is the variation coefficientfor the material property X.
Real design resistance, in contrast to the design val-ues of material or product properties, also includesuncertainties in the resistance model, e.g. geometricdeviations. In this case, the design resistance Rd isdefined by
Rd = 1
γRdR
(η
γmXk
)= Rk
γM, (8)
where γM is the partial safety factor covering the uncer-tainties in the resistance model (including the par-tial factor for the uncertainty in the material prop-erty described by γm, the uncertainty in the structuralmodel of the structuralmembers and the geometric data
defined by γRd and η representing the mean value ofthe conversion factor that takes into account volumeand scale effects, the effects of moisture and tempera-ture, etc.).
Since the design resistance is defined accordingto DIN EN 1990 Eurocode (2010), the mathematicalframework for determining the partial material safetyfactor is presented below. Following the simplifiedlevel I approach according to DIN EN 1990 Eurocode(2010), whereby at this point a suitable and meaning-fulmechanicalmodel is assumed, then the partial safetyfactor γM is computed by
γM = γRd · γm
η= 1
η·exp(αR ·β ·VR−1.645 ·VF), (9)
where αR represents a weighting factor for the resis-tance and β is the reliability index. The coefficient ofvariation VR is computed by:
VR =√V 2M + V 2
G + V 2F , (10)
where the components are defined as:
– VM: coefficient of variation for model uncertaintyof structural silicone sealant
– VG: coefficient of variation for the geometry– VF: coefficient of variation for the structural sili-cone sealant strength.
The coefficient of variation for the structural siliconesealant strength can be calculated from experimentaldata by
VF =√exp (s2X) − 1, (11)
where the standard deviation sX is given by
sX =√√√√ 1
n − 1
n∑i=1
(ln xi − mX)2. (12)
The standard deviation sX depends on the mean valuemX of the strength, which is computed via
mX = 1
n
n∑i=1
ln (xi ). (13)
Based on this and applying a log-normal distributionfor the material property X, which is strongly recom-
123
M. Drass, M. A. Kraus
Table 1 Values of kn for estimation of characteristic values (5% fractiles), (DIN EN 1990 Eurocode 2010)
n 5 8 10 20 ∞unknown VX 2.33 2.00 1.92 1.76 1.64
mended by Fischer (2001) for small sample sizeswith acertain coefficient of variation, (7) can be reformulatedby
Xk = exp (mX − kn · sX), (14)
which can be used to calculate the characteristic failurestrength of the analyses material, e.g. a DOWSIL 993structural silicone. The quantile factor kn is providedin Tab. 1.
Note that the partial safety factor alone is not suffi-cient for the design, but that the strength parameter isalso required for the design limit state analysis. Bothwill depend in value on the applied limit state functionand statistical modeling, which will be shown later inthis paper. In general, with increasing modeling andtesting efforts, both the partial safety factor and thecharacteristic strength value can be estimated with asmall uncertainty and thus smaller safety factors.
3.2 Partial material safety factor for DOWSIL 993using a stretch-based limit state function
In contrast to publications on the calibration of par-tial safety factors for stress based limit state functions,cf. Drass and Kraus (2020), Kraus and Drass (2020),the present work calibrates the partial safety factor forDOWSIL 993 based on experimental data from Staudtet al. (2018), Staudt (2017), Rosendahl et al. (2019)using an adaptive stretch-based failure criterion.
Returning to the computation of the partial safetyfactor γM for the structural silicone adhesive DOW-SIL 993 considering a stretch-based failure criterion(i.e. limit state), the proposed formula apparatus is com-bined with constraints from ETAG 002 (2012) to cal-culate γM using the Level I method of DIN EN 1990Eurocode (2010). Starting with the determination ofthe coefficients of variation for the stretch-based limitstate function, the coefficient for model uncertaintieswill be derived in the following.
3.2.1 Model uncertainties for the stretch-based limitstate function: coefficient of variation
The derivation of the model uncertainties for thepartial safety factor γM for the structural siliconeDOWSIL 993 is calibrated using measurement dataof different stretch failure modes based on the inves-tigations of Staudt et al. (2018), Staudt (2017) andRosendahl et al. (2019).
To describe the multiaxial failure of the siliconeadhesive, uniaxial tension and compression tests andso-called circular shear tests were performed by Staudt(2017). The details of the tests are described in Staudt(2017) and in Rosendahl et al. (2019) the experimen-tally determined failure points in the three-dimensionalstretch space of all individual tests were summarized.
In the one-dimensional case, the stretch λ is a mea-sure of the elongation or normal strain of a differentialline element. It is defined as the ratio between the finallength l and the initial length l0 of the material lineelement:
λ = l
l0. (15)
The relationship between the engineering or nominalstrain and the stretch is given by
ε = l − l0l0
= λ − 1. (16)
For the sake of clarity, a derivation of the stretch inthree-dimensional space is not shown here, but onlyreferred to in Appendix A.
In order to describe the model inaccuracies withregard to the failure description of DOWSIL 993 underarbitrary deformation, the three-dimensional failurestretches tabulated in Rosendahl et al. (2019) are trans-ferred to a two-dimensional space, the so-called π
plane. Again, the exact derivation of the π plane isomitted and reference is made to Appendix B. The rep-resentation of the failure points in the π plane is shownin Fig. 4a.
The aim is to approximate the failure stretches ofthe analysed structural silicone using a suitable failurecriterion. A very adaptable and comprehensible crite-rion for isochoric distortional failure was proposed byPodgórski (1984) and developed in a similar manner byBigoni and Piccolroaz (2004). In the following this cri-terion is abbreviated as PBP criterion. Using the nota-tion of Rosendahl et al. (2018), the distortional failurecriterion Φ iso is described in stretch space by
123
Dimensioning of Silicone Adhesive Joints...
Fig. 4 Representation ofthe failure stretches ofDOWSIL 993 in thetwo-dimensional π planeand (b) approximation ofthe failures stretches by thePBP and von Mises-likestretch-based failure criteria
Φ iso = λisoeq − λc == ρ cos
(β
π
6− 1
3arccos
(γ cos (3θ)
))
−λc = 0, (17)
where ρ represents the distance from the hydrostaticaxis to the boundary of the failure surface in the devi-atoric plane and θ is the so-called Lode angle (seeAppendix A). Both are functions of the deformationstate. The parameters β and γ determine the shape ofthe failure surface and must be determined based onexperiments. The threshold λc describes the size of thefailure surface. In order to guarantee a convex failuresurface, the shape parameters are restricted to the inter-vals β ∈ [0, 2] and γ ∈ [0, 1].
Approximating the failure stretches ofDOWSIL993with the PBP criterion results in the following two-dimensional failure surface, which is also shown in theπ plane in Fig. 4b. For the sake of completeness, theapproximation by the von Mises-like criterion
λc =√3I ′
2 (18)
is also shown, where I ′2 corresponds to the second
invariant of the deviatoric part of the Hencky straintensor.
As can be easily seen, the failure points for threedifferent deformation states, namely uniaxial tensionand compression and shear, are verywell approximatedwith the PBP criterion. In contrast, the von Mises-likecriterion is not at all suitable to describe the failurestates, which must lead to a very high model uncer-tainty. On this basis, themodel uncertainty is calculatedaccording to the explanations of Gulvanessian et al.(2012b). In relation to the PBP criterion, an equiva-
lent stretch of λc = 1.2538 was fitted, whereas that ofMises provides λc = 2.5691. If the model uncertaintyfor the PBP criterion is evaluated under the assump-tion of a normal distribution, a variation coefficient ofVM = 0.059 results. Assuming a lognormal distribu-tion, which is especially useful for small amounts ofdata, a VM = 0.0594 results. As a reminder, the nor-mal distribution is defined as
f (x) = 1
σ√2π
e− 1
2
(x−μσ
)2, (19)
where σ represents the standard deviation and μ givesus the mean value. In contrast, the lognormal distribu-tion function is defined by
f (x) = 1
σ x√2π
e− (ln(x)−μ)2
2σ2 , (20)
where σ defines the scale parameter and μ is the shapeparameter.
Accordingly, both distribution functions deliver analmost identical value, which is later used to calculatethe partial material safety factor.
3.2.2 Geometry uncertainties: coefficient of variation
Furthermore, the geometric deviations of the adhesivejoint must be taken into account to evaluate the partialsafety factor for DOWSIL 993. Since there is no exactknowledge of the geometry from the underlying dataset, as it was simply not measured exactly, an assump-tion must be made for the geometry uncertainty. Here,a value of VG = 0.10 is assumed. This guideline valueis based on personal communications and experiencesof the Seele company. Since the value of the geometric
123
M. Drass, M. A. Kraus
Fig. 5 Representation ofthe distribution functionsunder the assumption of a anormal distribution and blog-normal distribution forthe calculation of thecoefficient of variation ofthe model uncertainty,taking into account the PBPcriterion and the failurestretches
uncertainty is only an assumption, it can be adjustedindividually and reduced by factory production con-trols in the form of measurements and consecutive sta-tistical analysis. In summary, however, this value lieswithin a trustworthy range based on experience withindustrial applications of DOWSIL 993 (Fig. 5).
3.2.3 Determination of partial safety factor γM for astretch-based limit state function
In terms of (9), the weighting factor for the resistanceaccording to the Level I method is assumed to be αR =0.8. According to DIN EN 1990 Eurocode (2010) thisfactor is on the safe side. Generally, the reliability indexis considered with β = 3.8 for the ultimate limit state,which corresponds to a permanent design situationwitha target lifetime of the building of 50 years.
So far, the conversion factor η according to (9) hasnot been considered in the computations of the partialmaterial safety factor for the stretch-based limit statefunction. To account for further model uncertaintiesand conversion aspects, the conversion factor η is nowlinked to requirements fromETAG002 (2012) to have areasonable assumption regarding themodel uncertaintyunder consideration of ageing effects.
Typically, ageing phenomena occur in façades dueto water, temperature, UV, NaCl, SO2, detergent expo-sure. These adverse ageing effects are experimentallytested according to ETAG 002 (2012), where the ratioof the aged nominal strength (in terms of engineeringstress) to the unaged strengthmust be greater than 75%.Figure 6a shows the barrier in accordance to ETAG002 (2012) for tensile loading of an ETAG H-probe asa grey box according to the experimental engineering
stress data available. In addition to the box plot of thestrengths (engineering stresses), the log-normal distri-bution of the unaged and artificially aged tests is shownin Fig. 6b. It is interesting to see that the slopes changeat different temperatures. Accordingly, the temperaturehas a great influence on the distribution of the engineer-ing stress strengths. If one looks at the artificially agedsamples with NaCl and SO2, the gradient and thus thedistribution of the strengths changes only slightly.
All averages of the nominal strengths of the artifi-cially aged specimens are above the 75 % criterion,thus meeting the requirements of ETAG 002 (2012).This criterion therefore provides a lower limit valuewhich must be met experimentally in order to be ableto construct an SSG façade. Assuming this lower limitvalue is a true barrier according to ETAG 002 (2012),which includes all harmful influences such as temper-ature, water and UV storage as well as salt exposure,the conversion factor η can be determined accordingly:
η = 0.75. (21)
This is a reasonable approach, creates a lower boundaryfor η and links ETAG 002 concept with DIN EN 1990Eurocode (2010). It is to note that the conversion factorη can be adjusted according to the results of the ageingtests, if test results are available.
If one takes all previous assumptions as a basis andassuming that the uncertainties in the structural modelof structuralmembers is γRd = 1, then the partial safetyfactor γM for DOWSIL 993 with a stretch-based limitstate function reads
γM = 1.81. (22)
This value for the partial safety factors assumes 10 %for the coefficient of variation of the geometry and
123
Dimensioning of Silicone Adhesive Joints...
Fig. 6 a Box plot of nominal failure strengths of DOWSIL 993 under tensile loading and b lognormal distribution of the engineeringstress strengths of DOWSIL 993 after artificial ageing in tensile tests (cf. Drass and Kraus 2020; Kraus and Drass 2020)
VM = 0.0593 for the model VM. If the conversionfactor is taken into account, η = 0.75 must also beassumed. It is important to note that this partial safetyfactor is further adjustable by reducing the coefficientof variation of the adhesive joint geometry as a resultof accurate factorymonitoring andmachine applicationof the adhesive joint.
A very important point at this stage is the assump-tion that for the uncertainties of the structural modelof structural members the value is set to 1. This fac-tor will be adapted for simulation by FEM, as will beshown later, in order to avoid the problem of stress sin-gularities and thus make the FE solution independentof the mesh. The γRd factor will be determined by onesimulation of an ETAG H-sample and then used forthe calculation of the design resistance. The individualadaptation of γRd makes it possible to evaluate struc-tural components independent of the mesh.
3.3 Discussion of the determined partial safety factorfor structural silicone
Having obtained numerical value for the partial mate-rial safety factor with associated characteristic valuesfor the DOWSIL 993, this section discusses the under-lying assumptions in more detail.
In order to create direct comparability between thepartial safety factor according to EC0 and the globalsafety factor γtot according to ETAG 002 (2012), it isassumedwhen converting γM into a global safety factorthat only live loads affect the component. As a result,γM ismultiplied by the partial safety factor on the actionside of 1.5, resulting in a conservative global safetyfactor of
123
M. Drass, M. A. Kraus
γ ∗tot = γM · γEd = 1.81 · 1.5 = 2.72 � 6.0. (23)
Comparing this with the global safety factor accordingto ETAG 002 (2012), a large reduction results despiteconsideration of damaging influences. The principalassumptions for the application of the proposed con-cept or the determined partial safety factor, however, isthe application of suitable material models and failurecriteria for the structural silicone under investigationin order to have the lowest possible uncertainty on thematerial side. However, the following points allow formore detailed and precise computation of the partialmaterial safety factor in future research:
– ’realistic’ ageing protocols (deduction of load com-bination factors),
– fatigue,– viscoelasticity,– different performance / limit state functions
g(E, R),– multiple failure modes (distortional and/or dilata-tion) of the sealant,
– failure modes of the sealing application and thesealed system (series and/or parallel system behav-ior).
4 Validation example
The aim of this section is to show how to apply thesemi-probabilistic safety concept for the static proofof silicone-adhesive connections using the FEM. Thisexample deals in particular with a bonded façade ele-ment where the maximum load-bearing capacity underwind load is going to be calculated. Using the semi-probabilistic safety concept, the aim is to increase theapplied wind suction load until the load-bearing capac-ity of the structural silicone adhesive is reached. A spe-cial feature of this section is that the verification of theadhesive joint is to be carried out via FE simulations.This results in the peculiarity that in the simulationwith finite elements stress singularities at bi-materialnotches occur, whereby a mesh-dependent solution isobtained. This must be taken into account or circum-vented in the design approach which will be shownlater.
Since the partial safety factor on the basis of (22)has been calculated without considering uncertaintiesin the structural model of the structural members by
γRd,which in the following is equatedwith stress singu-larities at bi-material notches, the parameter γRd mustbe additionally calibrated to obtain amesh-independentsolution for the determination of the ultimate load. Toillustrate how this works, the verification concept andthe necessary steps are briefly introduced below andthen explained in detail using the example describedabove.
4.1 Methodology
The verification concept wil be divided into four steps,which are briefly described in the following:
Step 1: PBP criterion for 5% Quantile Values
– calculate 5% quantile values for uniaxial tension /compression and shear tests
– fit PBP criterion on 5% quantile values of experi-ments
– determine an equivalent stretch based on the 5%quantile Values → λc,5%
Step 2: Definition FE-Mesh
– definition of the FEmesh for the globalmodel of thefaçade element, i.e. the mesh density of the siliconeadhesive must be specified.
– definition of the FE-mesh for the ETAG H-Probe– note: both FE-meshes must be identical!
Step 3: Calculation of γRd
– calculation ETAG H-sample in tension with Fk,5%and predefined mesh from Step 2
– evaluation of the principal stretches in the FE-model
– fit PBP criterion → λc,Rd– calculation of γRd = λc,5%/λc,Rd
Step 4: Calculation of the design value λc,d
In the following steps 1-4 are presented in detail usingthe example of the bonded façade element located inBerlin, Germany. Details on the project will be pro-vided, when the numerical model will be explainedmore in detail.
4.1.1 Step 1: PBP criterion for 5% quantile values
As one can see in Fig. 4a, the individual experiments,here uniaxial tension/compression and shear, are scat-tering, so that onemust determine the 5%quantile value
123
Dimensioning of Silicone Adhesive Joints...
Fig. 7 Representation of the failure stretches of DOWSIL 993in the two-dimensional π plane and the approximation of thefailures stretches by the PBP-Experiment and the 5% quantilevalues
for each experiment individually on this basis. Hence,to calculate the 5% quantile values for the stretches inaccordance to ETAG 002 (2012), the following equa-tion can be utilized:
Ru,5% = Xmean − τα,β · sX. (24)
By calculating the 5%quantile values, the adaptivePBPfailure criterion must be fitted to these data in orderto calculate an equivalent stretch λc,5%. By doing thisyou get the equivalent stretch λc,5% = 1.09592. Forthe representation of the π plane this means that thefailure domain becomes slightly smaller (see Fig. 7).
4.1.2 Step 2: definition FE-mesh
In this section, a discretization for the global model ofthe façade elementmust be specified, which can also betransferred to the ETAG H-shaped test sample. Sincewe emphasize in this paper that we are able to calculatefailure loads or carrying loads respectively independentof the mesh, we will examine three mesh variants in thefollowing:
– 4 × 4 × 4 mm,– 3 × 3 × 3 mm,– 2 × 2 × 2 mm.
Ascanbe seen from the list, the silicone ismodeledwithbrick elements with exactly the same edge lengths. Allfollowing calculations are therefore carried out withthese three different discretizations. As an example ofthe mesh variant 2 × 2 × 2 mm, Fig. 8 shows theETAG H-shaped sample and a section of the façadeelement with the same mesh. The contact formulationin these examples was realized with so called Multi-
Fig. 8 Illustration of the FE-mesh of a theH-sample and b detailof the global model of the façade element including the glasspane, adhesive joint and aluminium profilewhere the colour blue
stands for glass, orange-red for the silicone adhesive and greyfor aluminium
123
M. Drass, M. A. Kraus
Point-Constrained contact elements, so that due to thisconstraint non coincident nodes must be used in themesh. However, it is essential that the contact formula-tion of the H-sample is identical to the formulation inthe global model.
4.1.3 Step 3: calculation of γRd
In the third step the uncertainty in the structural modelof structural members γRd will be determined. Theauthors are of the opinion that this safety factor shouldbe calibrated using the stress singularities occurring inFE calculations as input. In the context of FEM, stresssingularities are understood as the solution depends onthe mesh density in the area of bi-material notches.These notches always occur in the region of the stick-ing of the elements of the silicone with glass or the sub-structure. As a result, the stresses, strains and stretchesalways increase with finer mesh at the same load level.However, this effect must be avoided when dimension-ing silicone adhesive joints, as otherwise the solutiondepends on themesh, whichmeans that it is very lightlyto incorrectly dimension the joint. It is therefore of deci-sive importance to develop a method which, for exam-ple in the load bearing capacity calculation, alwaysleads to the same results without being dependent onthe mesh density.
Therefore, to be independent of the mesh density,γRd must be calibrated by one single numerical calcu-lation of an ETAG H-sample under tensile load. Theadvantages are obvious. On the one hand, this safetyfactor can be determined by one simple numerical cal-culation, and on the other hand, complex mathematicalmethods such as finite fracture mechanics and expen-sive fracturemechanics tests can be omitted. In order togive the reader a clear understanding of this procedure,all necessary steps for determining γRd are presentedin detail below.
As a first step, the geometry of the ETAG H-specimen or similar geometry must be entered into anFE program and meshed with the same mesh as that ofthe global model. This sample must then be pulled tothe 5% quantile value of the tensile strength or tensileforce. For the geometry of the H-shaped specimen of12× 12× 60 mm selected here, the tensile force to beapplied reads
Fk,5% = a · b · σdes · γtot
= 12mm · 60mm · 0.14 · 6 = 608.4N. (25)
Table 2 Simulation results for modified ETAG H-probe undertensile load for three different mesh densities and evaluation ofthe equivalent stretch according to PBP-criterion
mesh Force λ1 λ2 λ3 λc,Rd(mm) (N) (/) (/) (/) (/)
2 × 2 × 2 608.4 1.7351 0.9159 0.6298 0.5681
3 × 3 × 3 608.4 1.6214 0.9102 0.6782 0.4798
4 × 4 × 4 608.4 1.5548 0.9041 0.7121 0.4224
In this context, the pulled surface of the specimen(a · b) is multiplied by the design strength σdes ofDOWSIL 993 and the global safety factor of γtot = 6to determine the 5% quantile value of the tensile forceFk,5%. This determined force is applied to theH-shapednumerical examples with three different mesh densitiesand then the governing stretches in the corners are eval-uated. The mesh-dependent stretches are then in turnfitted by the PBP criterion again, so that three differentequivalent stretches λc,Rd are obtained for the three dif-ferent FE meshes. A summary is presented in Table 2.
The PBP-criterion is then fitted with the mesh-dependent failure stretches determined from finite ele-ment simulations. This results in corresponding equiv-alent stretches λc,Rd, which are also summarized inTable 2 and are additionally shown in the π plane plotin Fig. 9. What can be clearly seen is, on the one hand,that with increasingly finer mesh the failure surfacebecomes larger and, on the other hand, that the fail-ure surfaces calibrated on the FE calculations are sig-nificantly smaller than the 5% quantile failure surfacedetermined on the experimental tests on DOWSIL 993.
To calculate the uncertainty in the structural modelof the structural members, the following equation canbe adduced:
γRd = λc,5%
λc,Rd. (26)
The following table (see Table 3) summarizes this oper-ation for the three different meshes:
4.1.4 Calculation of the design value λc,d
In the last step, the determined partial safety factorfrom (22) and the partial safety factor for uncertaintyin the structural model of the structural members γRdmust be harmonizedwith the PBP-criterion, so that FE-
123
Dimensioning of Silicone Adhesive Joints...
Fig. 9 Representation of the failure stretches of the two-dimensional π plane and the PBP-criterion for the experiments, the 5% quantilevalues and the approximation of the failures stretches accordingly to Table 2 with respect to the mesh density
Table 3 Evaluation of the uncertainty in the structural model ofthe structural members γRd for three different mesh densities
Mesh λc,5% λc,Rd γRd(mm) (/) (/) (/)
2 × 2 × 2 1.0959 0.5681 1.9292
3 × 3 × 3 1.0959 0.4798 2.2842
4 × 4 × 4 1.0959 0.4224 2.5946
calculations can be carried out and carrying loads deter-mined without being dependent on the mesh.
This is done simply by dividing the equivalentstretch λc,5% by both partial safety factors, so that thefollowing applies
λc,d = λc,5%
γM · γRd. (27)
Since we examined three different mesh densities, theresults are summarized in the following table (seeTable 4).
It should be noted that with Table 4 and more pre-cisely the value for λc,d the dimensioning of the sil-icone adhesive joint can be carried out with the cor-responding mesh density. If one decides on a meshdensity of 2 × 2 × 2 mm for the structural siliconein the global model of the façade element, the valueλc,d = 0.3138 must be entered for the evaluation of
Table 4 Evaluation of the equivalent stretches λc,d for threedifferent mesh densities
Mesh λc,5% γM γRd λc,d(mm) (/) (/) (/) (/)
2 × 2 × 2 1.0959 1.81 1.9292 0.3138
3 × 3 × 3 1.0959 1.81 2.2842 0.2651
4 × 4 × 4 1.0959 1.81 2.5946 0.2333
the PBP criterion within the FE calculation. If this pro-cedure is followed, the silicone verification is carriedout for the first time using the semi-probabilistic safetyconcept accordingly to EN 1990 Eurocode (2002) andit is also possible to determine carrying loads that areindependent of themesh density. A last important pointis that with the tabulated values for λc,d any adhesivelybonded facade structures can be calculatedwith the cor-responding mesh density, as long as the correspondingtabulated value from Table 4 is used. Therefore, this isthe first mesh-independent approach to design bondedstructures using FEM.
4.2 Example: bonded façade element
In this example, a bonded façade element has to beverified statically according to EN 1990 Eurocode(2002), i.e. the semi-probabilistic safety concept. In
123
M. Drass, M. A. Kraus
Fig. 10 Two renderings ofthe project Voltair of thebuilding owner VOLTGmbH & Co. KG,Uhlandstraße 181-183,10623 Berlin
particular, this section deals with the numerical simu-lation of a bonded façade element with the dimensions2700×5100mm in order tomake the static proof of thestructural silicone adhesive. The product DOWSIL 993is fictionally used as adhesive in the present example,which has the following joint dimensions (28×12mm).A rendering of the bonded facade of the Voltair projectin Berlin is shown in Fig. 10.
In this example, the present concept from Sect. 3is used to determine the maximum load due to windsuction. Figure 11 shows the geometric model of thebonded façade construction exploiting symmetry.
The model consists of a laminated safety glass(LSG) 2 x 10 mm annealed glass with PVB layer inbetween. For reasons of simplicity, the LSG is repre-sented with a surrogate model, i.e. a mono-pane. This
mono-pane is then structurally bonded to the aluminumprofile with the silicone adhesive DOWSIL 993. Thebearing of the profile is done point-wise by so calledtoogle systems, so that the profile is kept at a distance of300mm.The external load is awind suction load,whichis increased until the maximum load-bearing capacityfor the silicone adhesive is reached.
4.2.1 ETAG 002: analytical approach
Starting with the analytical calculation of the prob-lem, ETAG 002 provides a hand calculation formula toobtain the maximum design wind load in accordanceto the ASD approach utilizing a global safety concept,which is computed via
pclassick,ETAG = 2 · σdes · hca
123
Dimensioning of Silicone Adhesive Joints...
Fig. 11 CAD model of abonded façade element witha = 5100 mm andb = 2700 mm underrepresentation of thebearing conditions and thewind suction load
= 2 · 0.14 · 282700
· 1000 = 2.91kN
m2 . (28)
The advantages of the mentioned approach are the sim-plicity and quick application, but the disadvantagesare the assumption of linear elastic material and struc-tural behavior, the applicationof engineering stresses asdesign basis, the separation of tensile and shear stressesdue to different actions, the high safety level, the verystrong simplification of the load transfer, the neglectof uncertainties on the action side and the separateverification of bonded facade elements, since differ-ent safety concepts are used for glass and silicone. Dueto the numerous disadvantages, it is essential to offermodern dimensioning approaches that circumvent theabove mentioned disadvantages and still allow a safedesign of the adhesive joint and the structure. In addi-tion, FEM verification is often required in projects, asthe simplifications according to ETAG are too blatantand building owners and engineers are uncertain aboutits application. Therefore, the authors are of the opinionthat for modern, bonded facades, the ETAG approachcan be used for the basic evaluation and a preliminarycalculation of the ultimate load, but that a more precise
verification by means of FEM is decisive, since due toincreasingly complex building structures and bondedjoint geometries, the verification of the bond is cer-tainly more complex than is covered by ETAG.
4.2.2 Finite element analyses
In this section, the results of the FE calculations arebriefly presented. As the model is non-linear, the cal-culations are performed under large deformations. Inorder to obtain a good convergence, the wind suc-tion load is successively increased until the PBP cri-terion in the silicone is exceeded. It is important tonote that three individual calculations of the façade ele-ment were carried out, each with a different mesh den-sity of the silicone. The mesh density was previouslyselected according to Sect. 4.1.2. The PBP criterionis also adjusted during its evaluation according to theimplemented mesh density, so that for λc,d the valuesaccording to Table 4 must be used.
Figure 12gives a histogramshowing the numericallydetermined load capacities for different mesh densi-ties. It can clearly be seen that the maximum wind
123
M. Drass, M. A. Kraus
Fig. 12 Histogram ofpermissible wind loads withrespect to the chosen safetyconcept:, i.e Finite ElementAnalysis-Limit State Designanalyzing three differentmesh densities (FEA-LSD)with semi-probabilisticsafety concept andallowable stress designmethod with global safetyconcept in accordance toETAG 002 (2012)(ETAG-ASD)
suction load that can be absorbed according to EN1990 Eurocode (2002) design reads pd = 4.08 kN/m2.It should also be noted that with the proposed con-cept from Sect. 4.1, the same loads could be deter-mined for three different mesh densities. This meansthat the solution with the method proposed is indepen-dent of the meshing. This is a very important resultbecause the concept is easily accessible, can be cal-ibrated by engineers and complex methods such asfinite fracture mechanics are not required. As can beseen additionally from the histogram, the wind suctionload that can be absorbed based on ETAG’s approachis pclassick,ETAG = 2.91 kN/m2 using (28).
Here, it must be noted that wind loads that can beabsorbed accordingly to EN 1990 Eurocode (2002)are so-called design loads, i.e. according to EN 1990Eurocode (2002) a factor of 1.5 is still included on theaction side.
In order to compare the wind load determinedaccording to ETAG 002 (2012) with the wind loadsmentioned above, the value according to ETAG maybe modified as follows:
pclassicd,ETAG = 1.5 · pclassick,ETAG
= 1.5 · 2.91 = 4.37 kN/m2.(29)
The comparison between (29) with the ultimateloads based on the FEM and semi-probabilistic safetyconcept is actually not quite correct, since, accord-ing to ETAG’s calculation method, one has alreadyfound its limit state function, i.e. no more wind loadmay be applied to the bonded facade. Nevertheless,we compare the value of ETAG with the value accord-ing to Eurocode by the fictitious increase of the cal-culated wind suction load from (28) with the factor of
γQ = 1.5. However, a very interesting result is thatboth approaches calculate almost identical maximumloads. However, since according to ETAG a slightlyhigher wind load was determined, this approach is onthe unsafe side according to the considerations fromthe Eurocode.
5 Discussion and conclusions
This paper deals with the presentation and calcula-tion of a partial safety factor for the structural siliconeDOWSIL 993. A semi-probabilistic approach accord-ing to EN 1990 Eurocode (2002) was proposed andapplied to determine a suitable partial safety factor.To illustrate the procedure, the methodological outlineconcludes with an exemplary probabilistic evaluationof a specific limit state for the structural silicone adhe-sive DOWSIL 993. The methodology is state of theart, but was applied for the first time to structural sili-cone adhesives with the application area of the façade.Furthermore, the concept according to the EN 1990Eurocode (2002) was extended to the finite elementmethod in such a way that a design is now possiblewithout obtaining mesh-dependent results.
The value determined within the scope of this workshows that the partial material safety factor (also tak-ing into account temperature and ageing and laboratoryeffects) is γM = 1.81 and thus a global partial safetyfactor of γ ∗
tot = 2.72 is justified, in the case of precisematerial modeling and failure description via the PBPcriterion. Furthermore, the semi-probabilistic conceptwas extended to the finite element method in such away that it is now possible for the first time to carryout Eurocode-compliant and even mesh-independentdimensioning of silicone adhesives in the façade sector.
123
Dimensioning of Silicone Adhesive Joints...
In conclusion, however, it should be clearly statedthat the concept presented and its application are sub-ject to the following conditions:
– Application of a hyperelastic material model thatcan exactly represent any deformation state, e.g.Drass et al. (2018), Drass et al. (2019a) .
– The use of the stretch-based PBP failure criterionis mandatory.
– For now, only applicable to isochore failure.– All given values are only applicable for the productDOWSIL 993.
Acknowledgements Open Access funding provided by Pro-jekt DEAL. The authors would like to thank the people andcompanies that contributed to this publication by making data,projects and measurements available. Special thanks go to thebuilding owner VOLTGmbH&Co. KG, Uhlandstraße 181-183,10623 Berlin as well as the company Knippers Helbig, FlorianScheible and Thiemo Fildhuth for providing us the renderingand project details the Voltair bulding in Berlin, Germany. Fur-thermore the authors would like to thank Schütz GoldschmidtSchneider Ingenieurdienstleistungen im Bauwesen GmbH andespecially Sebastian Schula to be involved in the Voltair project.
Open Access This article is licensed under a Creative Com-mons Attribution 4.0 International License, which permits use,sharing, adaptation, distribution and reproduction in anymediumor format, as long as you give appropriate credit to the originalauthor(s) and the source, provide a link to the Creative Com-mons licence, and indicate if changes were made. The images orother third partymaterial in this article are included in the article’sCreative Commons licence, unless indicated otherwise in a creditline to thematerial. If material is not included in the article’s Cre-ative Commons licence and your intended use is not permitted bystatutory regulation or exceeds the permitted use, you will needto obtain permission directly from the copyright holder. To viewa copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
A Measures of deformation
The deformation gradient F maps a material line ele-ment dX from the reference configuration to the corre-sponding line element in the current configuration dx:
dx = F dX, (A.30)
where
F = ∂x∂X
= ∇Xx. (A.31)
This second-order tensor is defined as a two-point ten-sor because it refers to the reference configuration and
the current configuration. Its determinant
J = det F, (A.32)
characterizes volume change. Splitting the deformationgradient into a rotation tensor and a stretch tensor, thetheorem of polar decomposition yields
F = RU = vR, (A.33)
where R = RT is an orthogonal rotation tensor, Uthe right stretch tensor and v the left stretch tensor.The right stretch tensor U is also known as Lagrangianstretch tensor and the left stretch tensor v as Eulerianstretch tensor based on their corresponding underlyingconfiguration. Both stretch tensors can be also given inspectral representation reading
U =3∑
i=1
λi (Ni ⊗ Ni ) and v =3∑
i=1
λi (ni ⊗ ni ) ,
(A.34)
where λi represent the principal stretches (eigenval-ues) and Ni and ni define the eigenvectors of U and v,respectively. The principal invariants of, e.g., the leftstretch tensor v, read
Iv = tr (v) , IIv = 1
2
[I 2v − tr
(v2
)],
IIIv = det (v) . (A.35)
Let us denote the surface which represents theboundary between an intact and a damaged materialfailure surface Φ. A general stretch-based failure cri-terion is then given by
Φ (v) = 0. (A.36)
In general, the material does not fail for Φ (v) < 0.Φ (v) = 0 (and hypotheticallyΦ (v) > 0) correspondsto failure. In this context it is important to note thatmaterial failuremay correspond to yielding, stress soft-ening effects or crack nucleation which not necessarilyrepresents the ultimate failure of thematerial. Owing toisotropy, failure criteria must be invariant with respectto arbitrary rotations of the coordinate system. Hence,criteria may be formulated in terms of principal invari-ants of the left stretch tensor. The trace of the left stretchtensor
123
M. Drass, M. A. Kraus
Iv = tr (v) = λ1 + λ2 + λ3. (A.37)
is related to the hydrostatic deformations and thereforeimportant for dilatational failure. Concerning distor-tional failure, the second and third invariants of thestress deviator are often used. Denoting invariants ofthe deviator with a prime, the second deviator invariantis given by
II ′v = 1
6
[(λ1 − λ2)
2 + (λ2 − λ3)2 + (λ3 − λ1)
2],
(A.38)
and the third invariant of the left stretch tensor deviatorcan be expressed as
III ′v =
(λ1 − Iv
3
) (λ2 − Iv
3
)(λ3 − Iv
3
). (A.39)
Failure criteria can be visualized using three-dimensio-nal or two-dimensional implicit plots. The sectionalplane of the failure surface with the deviatoric plane(also know as deviatoric plane) is often of great interest,especially with regard to the verification of convexity.Introducing a new orthogonal coordinate system withthe coordinates ξ1, ξ2 and ξ3, the deviator plane is char-acterized by ξ2 and ξ3, whereas the third coordinate ξ1is perpendicular to that plane and points in the directionof the hydrostatic axis (Schreyer 1989). The orthogonaltransformed coordinates read
ξ1 = λ1 + λ2 + λ3√3
, (A.40)
ξ2 = λ1 − λ3√2
, (A.41)
ξ3 = 2λ2 − λ1 − λ3√6
. (A.42)
The coordinate transformation between the princi-pal stretches and the transformed coordinates is also
expressed by
⎛⎝λ1
λ2λ3
⎞⎠ =
⎛⎜⎝
1√3
1√2
− 1√6
1√3
0 2√6
1√3
− 1√2
− 1√6
⎞⎟⎠
⎛⎝ξ1
ξ2ξ3
⎞⎠ . (A.43)
B Description of failure surfaces
A generic example for an illustration of a failure crite-rion in principal stretch space is given in Fig. 13. For abetter understanding, the transformed coordinate sys-tem and three deviatoric planes at different sectionalplanes are illustrated. Additionally, important meridi-ans for stress angles of θ = 0◦, 30◦, 60◦ are shown.These meridians are important for the parametriza-tion of failure criteria based on experimental or vir-tual datasets (Fahlbusch et al. 2016). In contrast to theclassical proposed invariants of Eqs. A.37–A.39, moredescriptive invariants with geometrical meaning wereintroduced by Novozhilov (Kolupaev 2018). They aredefined by the scaled hydrostatic axis ξ1, the distancebetween the failure surface to the hydrostatic axis ρ
and the stress angle θ . The radius ρ and the argumentθ of the stress angle cos 3θ are defined by
ρ =√
ξ22 + ξ23 = √2II ′
v (B.44)
and
θ = 1
3arccos
(3√3
2
III ′v(
II ′v)3/2
)with θ ∈ [0, π/3] .
(B.45)
Fig. 13 Haigh andWestergaard-space ofarbitrary cavitation criterionin terms of principal Cauchystresses (λ1, λ2, λ3), thetransformed coordinates ξ1,
ξ2 and ξ3 and the deviatoricplane at different sectionalplanes (α I1, β I1, γ I1)
123
Dimensioning of Silicone Adhesive Joints...
The so-calledmeridian plane, inwhich differentmerid-ians are represented in the coordinates ξ1 − ρ, is oftenused to illustrate three-dimensional failure criteria inas two-dimensional section planes (Zyczkowski 1981).Kolupaev (2018) recommends to scale the abscissa ofthe meridian plane with respect to the von Mises crite-rion such that the scaledmeridian plane is formulated in(Iv,
√3II ′
v) coordinates. For all following studies, thedefinition in accordance with Kolupaev (2018) will beapplied for illustrating the scaled meridian plane. Thevariable ϕ, which represents a variation of the stressangle, is defined accordingly to Kolupaev (2018).
References
ASTMC1401. (2002) Standard guide for structural sealant glaz-ing (2002)
Bedon, C., Santarsiero, M.: Transparency in structural glass sys-tems via mechanical, adhesive, and laminated connections -existing research anddevelopments.Adv.Eng.Mater.20(5),1700815 (2018). https://doi.org/10.1002/adem.201700815
Bigoni, D., Piccolroaz, A.: Yield criteria for quasibrittle and fric-tional materials. Int. J. Solids Struct. 41(11), 2855–2878(2004). https://doi.org/10.1016/j.ijsolstr.2003.12.024
Blockley, D.I.: Engineering Safety. McGraw-Hill, New York(1992)
Botz, M., Wilhelm, K., Siebert, G.: Experimental investigationson the creep behaviour of PVB under different temperaturesand humidity conditions. Glass Struct. Eng. 4(3), 389–402(2019). https://doi.org/10.1007/s40940-019-00098-2
DIN EN 1990 Eurocode (2010) Grundlagen der tragwerkspla-nung. Deutsche Fassung EN 1990: 2002+ Al: 2005+ A1:2005/AC: 2010 (2010)
Drass,M., Schwind, G., Schneider, J., Kolling, S.: Adhesive con-nections in glass structures—part I: experiments and ana-lytics on thin structural silicone. Glass Struct. Eng. 3(1),39–54 (2018). https://doi.org/10.1007/s40940-017-0046-5
Drass, M., Bartels, N., Schneider, J., Klein, D.: Pseudo-elasticcavitation model—part II: extension to cyclic behavior oftransparent silicone adhesives. Glass Struct. Eng. (2019a)
Drass,M.,DuBois, PA., Schneider, J.,Kolling, S.: Pseudo-elasticcavitation model—part I: finite element analyses on thinsilicone adhesives in façades. Glass Struct. Eng. (2019b)
Drass, M.A., Kraus, M.: Semi-probabilistic calibration of a par-tial material safety factor for structural silicone adhesives—part I: derivation. Int. J. Struct. Glass Adv. Mater. Res.4(2020), 56–68 (2020). https://doi.org/10.3844/sgamrsp.2020.56.68
EN 1990 Eurocode—basis of structural design (2002)ETAG 002 Guideline for european technical approval for struc-
tural sealantglazing kits (2012)Fahlbusch, N.C., Kolupaev, V.A., Becker, W.: Generalized Limit
Surfaces—With an Example of Hard Foams, pp. 337–365.Springer, Berlin (2016)
Fischer, L.: Das neue Sicherheitskonzept im Bauwesen: einLeitfaden für Bauingenieure, Architekten und Studen-
ten. Bautechnik : Spezial, Ernst. https://books.google.de/books?id=lxoLmwEACAAJ (2001)
Gulvanessian, H., Calgaro, JA., Holicky, M.: Designers’ guideto Eurocode: basis of structural design (2012a)
Gulvanessian, H., Calgaro, JA., Holicky, M.: Designers’ guideto eurocode: basis of structural design (2012b)
Kolupaev, V.A.: Equivalent Stress Concept for Limit State Anal-ysis, vol. 86. Springer, Berlin (2018). https://doi.org/10.1007/978-3-319-73049-3
Kraus, M.A., Drass, M.: Semi-probabilistic calibration of a par-tial material safety factor for structural silicone adhesives—part II: verification concept. Int. J. Struct. Glass Adv.Mater. Res.4(2020), 10–23 (2020). https://doi.org/10.3844/sgamrsp.2020.10.23
Kraus, M.A., Schuster, M., Kuntsche, J., Siebert, G., Schneider,J.: Parameter identification methods for visco- and hyper-elastic material models. Glass Struct. Eng. 2(2), 147–167(2017). https://doi.org/10.1007/s40940-017-0042-9
Marek, P., Kvedaras, A.: From partial factors design to fullyprobabilistic reliability assessment of structures. Statyba4(4), 252–259 (1998). https://doi.org/10.1080/13921525.1998.10531414
Podgórski, J.: Limit state condition and the dissipation functionfor isotropic materials. Arch. Mech. 36(3), 323–342 (1984)
Rosendahl, P.L., Drass, M., Schneider, J., Becker, W.: Cracknucleation in hyperelastic adhesive bonds. Eng. Transpar.2(5–6), 409–425 (2018). https://doi.org/10.1002/cepa.941
Rosendahl, P.L., Drass, M., Felger, J., Schneider, J., Becker, W.:Equivalent strain failure criterion for multiaxially loadedincompressible hyperelastic elastomers. Int. J. Solids Struct.166, 32–46 (2019). https://doi.org/10.1016/j.ijsolstr.2019.01.030
Santarsiero, M., Louter, C.: Metal-to-glass bond strength ofstructural PVB. In: GPD Glass Performance Days 2019—Proceedings (2019)
Schreyer, H.L.: Smooth limit surfaces for metals, con-crete, and geotechnical materials. J. Eng. Mech.115(9), 1960–1975 (1989). https://doi.org/10.1061/(ASCE)0733-9399(1989)115:9
Sørensen, JD.: Calibration of partial safety factors in danishstructural codes. In: JCSS Workshop on Reliability BasedCode Calibration, Citeseer, vol 21, p 2002 (2002)
Staudt, Y.: Proposal of a failure criterion of adhesively bondedconnections with silicone. Doctoral thesis, University ofLuxembourg (2017)
Staudt, Y., Odenbreit, C., Schneider, J.: Failure behaviour of sili-cone adhesive in bonded connections with simple geometry.Int. J. Adhes. Adhes. 82, 126–138 (2018). https://doi.org/10.1016/j.ijadhadh.2017.12.015
Zyczkowski,M.: Combined Loadings in the Theory of Plasticity.PWN-Polish Scientific Publ, Warszawa (1981)
Publisher’s Note Springer Nature remains neutral with regardto jurisdictional claims in published maps and institutional affil-iations.
123